Isovector and isoscalar pairing in low-lying states of N = Z nuclei GuanJian Fu Tongji University Jun. 5, 2017 “Advances in theoretical nuclear physics”, Stockholm

Nucleon pair: isospin singlet / triplet A proton and a neutron are the same kind of nucleons, with different z-components of isospin T = 1/2, Tz = +1/2 for = -1/2 for Nuclear force is charge independent approximately The Coulomb interaction doesn’t play an important role in low-lying states Isospin conservation singlet states and triplet states in isospin space, we have T = 1, Tz = 1 T = 1, Tz = -1 T = 1, Tz = 0 T = 0, Tz = 0 isospin triplet states isovector isospin singlet state isoscalar

Interactions with T = 1 attractive and short ranged: 𝛿 interaction 04:12 Interactions with T = 1 attractive and short ranged: 𝛿 interaction 𝑗𝑗𝐽𝑇 𝑉 12 (𝛿) 𝑗𝑗𝐽𝑇 R. F. Casten, Nuclear Structure from a Simple Perspective.

multi-j shell space + effective interaction Isovector J = 0 pairing multi-j shell space + effective interaction (generalized) seniority scheme: collective (correlated) J = 0 pair ground states of even-even semimagic nuclei are well described by S pair wave function! overlap > 0.92

Interactions with T = 0 4 + 2 + 5 + 3 + 7 + 1 + 0 + 04:12 Interactions with T = 0 attractive and short ranged: 𝛿 interaction 42Sc ( j = 7/2 ) 0 + 1 + 7 + 3 + 2 + 4 + 5 + 𝑗𝑗𝐽𝑇 𝑉 12 (𝛿) 𝑗𝑗𝐽𝑇 effective int. The question is whether or not a pairing correlated state or a pair condensation can be formed. R. F. Casten, Nuclear Structure from a Simple Perspective. S. Frauendorf and A. O. Macchiavelli, Prog. Part. Nucl. Phys 78, 24

Isoscalar spin-aligned pair of the g9/2 orbit

Isoscalar spin-aligned pair of the g9/2 orbit Single g9/2 shell calculation for 96Cd, the spin-aligned pair: pair condensation: |𝜑 = ( 𝐴 9,0 † × 𝐴 9,0 † ) (𝐼,0) |0 C. Qi, J. Blomqvist, T. Back, B. Cederwall, A. Johnson, R. J. Liotta, and R. Wyss, Phys. Scr. T 150, 014031. P. van Isacker, Int. J. Mod. Phys. E 22, 1330028.

Isoscalar spin-aligned pair of the g9/2 orbit Single g9/2 shell calculation for 92Pd, the spin-aligned pair: pair condensation basis: ( ( ( 𝐴 9,0 † ×𝐴 9,0 † ) ( 𝐽 2 ,0) ×𝐴 9,0 † ) ( 𝐽 3 ,0) ×𝐴 9,0 † ) (𝐼,0) |0

NPA: a pair truncation model for multi-j shells 04:12 NPA: a pair truncation model for multi-j shells Shell model, the “exact” many-body solution Nucleon-pair approximation (NPA) with isospin the pair basis: 关于配对波函数或者基于配对基矢的研究非常少 Calculate NPA matrix elements very complicated Pair truncation: the building block is (J,T) pairs Validity: overlap between pair (NPA) wave functions and SM wave functions in many-j shell calculation G. J. Fu, Y. Lei, Y. M. Zhao, S. Pittel, A. Arima, Phys. Rev. C 87, 044310 (2013). Y. M. Zhao and A. Arima, Phys. Rep. 545, 1 (2014)

96Cd: dual description S pair: J=0,T=1 D pair : J=2,T=1 K pair : J=8,T=1 A(9) pair : J=9,T=0 + JUN45 Dual description: Non-orthogonality feature of nucleon-pair basis! G. J. Fu, J. J. Shen, Y. M. Zhao, and A. Arima, Phys. Rev. C 87, 044312 (2013).

g.s. of N=Z even-even nuclei 04:12 g.s. of N=Z even-even nuclei Study N = Z even-even nuclei: many-j shells + effective interaction 20Ne, 24Mg, 32S, 36Ar : USDB 44Ti, 48Cr : GXPF1 60Zn, 64Ge, 92Pd, 96Cd : JUN45 For N = Z nuclei: G. J. Fu, Y. M. Zhao, and A. Arima, Phys. Rev. C 91, 054322 (2015)

g.s. of N=Z even-even nuclei 04:12 g.s. of N=Z even-even nuclei Study N = Z even-even nuclei: many-j shells + effective interaction 20Ne, 24Mg, 32S, 36Ar : USDB 44Ti, 48Cr : GXPF1 60Zn, 64Ge, 92Pd, 96Cd : JUN45 For N = Z nuclei: G. J. Fu, Y. M. Zhao, and A. Arima, Phys. Rev. C 91, 054322 (2015)

g.s. of N=Z even-even nuclei 04:12 g.s. of N=Z even-even nuclei Study N = Z even-even nuclei: many-j shells + effective interaction 20Ne, 24Mg, 32S, 36Ar : USDB 44Ti, 48Cr : GXPF1 60Zn, 64Ge, 92Pd, 96Cd : JUN45 For N = Z nuclei: G. J. Fu, Y. M. Zhao, and A. Arima, Phys. Rev. C 91, 054322 (2015)

g.s. of N=Z even-even nuclei 04:12 g.s. of N=Z even-even nuclei Study N = Z even-even nuclei: many-j shells + effective interaction 20Ne, 24Mg, 32S, 36Ar : USDB 44Ti, 48Cr : GXPF1 60Zn, 64Ge, 92Pd, 96Cd : JUN45 For N = Z nuclei: G. J. Fu, Y. M. Zhao, and A. Arima, Phys. Rev. C 91, 054322 (2015)

g.s. of N=Z even-even nuclei 04:12 g.s. of N=Z even-even nuclei Study N = Z even-even nuclei: many-j shells + effective interaction 20Ne, 24Mg, 32S, 36Ar : USDB 44Ti, 48Cr : GXPF1 60Zn, 64Ge, 92Pd, 96Cd : JUN45 For N = Z nuclei: For semimagic nuclei: (> 0.92) G. J. Fu, Y. M. Zhao, and A. Arima, Phys. Rev. C 91, 054322 (2015)

T = 0 states of N=Z odd-odd nuclei 04:12 T = 0 states of N=Z odd-odd nuclei Study low-lying T = 0 states of 3 valence protons and 3 valence neutrons: many-j shells + effective interaction 3 isovector pairs cannot form T = 0 states, have to break 1 pair; isoscalar pair condensation may be possible 22Na, 34Cl: USDB 46V: GXPF1 62Ga, 94Ag: JUN45

T = 0 states of N=Z odd-odd nuclei 04:12 T = 0 states of N=Z odd-odd nuclei Study low-lying T = 0 states of 3 valence protons and 3 valence neutrons: many-j shells + effective interaction 3 isovector pairs cannot form T = 0 states, have to break 1 pair; isoscalar pair condensation may be possible 22Na, 34Cl: USDB 46V: GXPF1 62Ga, 94Ag: JUN45 Preliminary results: J = 1, T = 0 pairing for the 1+ states, overlap is 0.02-0.49 for the 3+ states, overlap is 0.0001-0.59 J = 3, T = 0 pairing for the 9+ state of 34Cl and 62Ga, overlap > 0.92 J = 5, T = 0 pairing for the 15+ state of 46V, overlap > 0.99

Thanks for your attentions！ 04:12 Thanks for your attentions！

Charge independence W. Benenson and E. Kashy, Rev. Mod. Phys. 51, 527 (1979).

Charge independence D. D. Warner, M. A. Bentley, and P. van Isacker, Nat. Phys. 2, 311 (2006).

multi-j shell space + effective interaction Isovector J = 0 pairing (generalized) seniority scheme: collective (correlated) J = 0 pair ground states of even-even semimagic nuclei are well described by S pair wave function! multi-j shell space + effective interaction overlap > 0.92

Isovector J = 0 pairing overlap > 0.92 (generalized) seniority scheme: collective (correlated) J = 0 pair ground states of even-even semimagic nuclei are well described by S pair wave function! ground states of odd-mass semimagic nuclei: overlap > 0.92 V. Zelevinsky and A. Volya, Physics of Atomic Nuclei.